lol obv on that specific example, on the example here I think you are wrong b/c we lose more equity with the flop call and we cannot call correctly the turn. I'm sure stinkypete will love this thread once he comes back from the deads.

Now I'm really confused; someone please page stinkypete, I do remember his involvement in the AIEV vs SBSEV thread and he sure seemed to know what he was talking about.

Paging Tom Chambers as well.

I really don't want to be a dick here but you are simply dead wrong. You can do the math if you want but it's tedious so I'm not going to bother. Here's how you go about it.

There are 43 unknown cards that can come on the turn. We will fold on 8 of them. So 8/43 we will fold the turn and lose $8.10. Figure out our average equity on the 35/43 other cards. Multiply that by the final pot size ($67) and subtract our contribution ($29.45). Final calculation looks like this:

((35/43 * average equity on turns we call * $67) - $29.45) - (8/43 * $8.10) = EV of calling flop, calling/folding turn

Contrast this with ((equity on flop * $67) - $29.45) = EV of shoving flop.

The first number will be higher. Promise.

If you know his hand, and know his future action, it is ALWAYS better to call and see more streets. Proof: obv way too lazy... but, you have perfect information except for future cards, so you can play perfectly by calling. Note- unrealistic premises for this problem.

Similar hands have been talked about many many times.

In this thread a lot of players chimed in and DJ sensei pushed some numbers.

http://forumserver.twoplustwo.com/19...ml#post4849332

That's what I had come to semi-understand, but I can't clearly see it, as in I have no doubt it instinctively makes perfect sense to me, you know what I mean?

Also, if villain has a PF 3bet range of 1.5% and we know he's never getting away from AA** with less than a PSB left on the turn, I don't see it being that unrealisitic of a scenario.
Lots of bad players fit that description.

And this is mainly about the math and the theory behind it, some kind of solid framework with which to work on different scenarii.
Thought it was obvious to most players, but seeing the responses in the thread, I think it's far from it and is why I'd really like to see a specialist of this kind of stuff demonstrate it mathematically/direct me to a thread that already addresses this topic.

don't get me wrong, I like the premises, I was just noting that conclusion only holds in very specific cases.

In terms of understanding why call is better intuitively. Think about it like this: he is all in on the flop, you can put your money all in now, or, wait and see if it's A72. If it's one of these turns, then you can make the correct play and fold. This is the scenario in which the strategy of call flop dominates the strategy of allin on flop.
Now, at a certain stack depth, you will have to call all rivers, so allin flop becomes = call flop call turn. E.g., if pots 40, he bets 40 and you have 45 behind, it doesn't matter what you do, because you won't be able to fold any rivers correctly.

All right never mind all this. The thing is the "equity" op gave is just plain bad wrong. I used his numbers w/o crunching it. We cannot use the equity given for AA** vs our hand for all different turns because of all the times there is a split pot. I think you are right spaddle. I took all turn combination and our equity is 57.5% instead of the 64% needed. Not sure it would add up to 7% more in ev but w/e.

ProPokerTools Omaha Hi Simulation

600,000 trials (Randomized)

board: 287

Hand	Pot equity	Wins	Ties
AA**	50.50%	301,838	2,328
qs8sjh9d	49.50%	295,834	2,328

Using this flop/hands combo, I took every combination we should call on the turn and the equity is 57.46%. Using what you wrote, we have for flop call + turn call/fold

((35/43 * 57.46% * $67) - $29.45) - (8/43 * $8.10) = 0.4$

If we ship flop we have

67*.495 -29.45 = 3.715$

To have a higher ev calling flop/turn we would need on avg ~64% eq when we call the turn. I tried every possible turn in PPT and got that average which is far from what we need for the numbers to reflect what you said. Maybe I'm just super ******ed and I'm missing something super obvious here. Can someone point it out. Can the split pot add the 7% missing ?

lol, you're not ******ed, I am. The formula I gave you is wrong, sorry for not double-checking it and any confusion that may have caused. The correct formula is:

35/43 * (average equity on turns we call * $67 - $29.45) - (8/43 * $8.10) = EV of calling flop, calling/folding turn

Plugging in 57.46% average equity on the turn, we see that calling flop, calling/folding turn has an EV of ~$5.85, which is more than $2 over the EV of getting it in on the flop.

Villain has 3333 on KJ8ss flop and you have 2222. He bets half of your stack on the flop.

I think people are mistaken about the concept of 'losing equity' on bad turn cards. You don't magically 'lose equity' on turn cards - bad turn cards are already built in to your flop equity, and they are just an instance of already calculated negative variation. If you have 50% of flop equity, the weighted average of your turn equities has to also be 50%.

If your villain is bet/calling flop, and shoving any turn, he is for all practical reasons all-in from the flop onwards (and getting it in is better than folding OTF). The question is whether you can gain EV by folding some turns correctly, and intuitively the answer is yes. From the flop onwards, shoving flop has the same EV as calling flop and calling any turn, if the premise that villain has 100% AA, calls 100% on flop and shoves 100% on turn. And since calling turn is a mistake in some cases, folding has to result in better EV.

A simplified calculation (but not proof) could be a situation where you have 50% on flop, 25% on half turns, and 75% on the other half of turns. Let's say the pot is 1 chip and effective stack remaining is 4 chips. Calculating from flop onwards, getting it in right there is 9/2 - 4 = +0.5 sklanskychips, and calling flop to play turns perfect is 0.5(-1) + 0.5(0.75 * 9 - 4) = +0.875.

The calculation necessary to generalise the example to constitute valid proof would be very complicated, but follows a similar pattern.

Another thing that just came to mind, about the dilemma of flop to call with 34% of equity required, it seems to be a negative proposition calculating from the flop (since if making a flop play strategy, you need to include the flop bet as an investment), but, calling flop to fold 34% on the turn is a worse play, since you fold out positive expectation.

The real thing to understand about the situation anyways is that being able to make correct decisions, and gaining information anywhere in a hand results in gaining EV. Finding a situation that disproves that, would disprove the entire notion of expected value and equity that is widely accepted as the basis of poker (or even general gambling) theory.

I think the following is a formal proof that OP wanted. I tried to make it as clear and readable as possible.

We consider two strategies:
1) all-in: go all-in on flop. Assumption is opp. calls with everything (no fold equity).
2) decide-turn: call flop, only call all-in on turn if EV>=0

We want show EV(decide-turn) >= EV(all-in), assuming opp. always goes all-in on turn.

[Proof]
All-in strategy has the same EV as calling on the flop and calling on every turn. So:

EV(all-in) = (Sum_i EV(call-turn,turn=i) * Prob(turn=i)) - flop_call_size,
where:
- Sum_i is the sum over all possible turn cards, i denotes the turn card,
- EV(call-turn,turn=i) is the EV on turn for calling when turn card is i.

The decide-turn strategy only continues on turns with positive EV:

EV(decide-turn) = (Sum_i max(0, EV(call-turn,turn=i)) * Prob(turn=i)) - flop_call_size,
where:
- max(0, EV(call-turn,turn=i)) means that we continue on turns where EV>=0 and otherwise EV=0, which is the EV of folding.

The elements of the sums of EV(all-in) and EV(decide-turn) are the same except that all negative elements in EV(all-in) are replaced by 0 in EV(decide-turn).

Thus, EV(decide-turn) >= EV(all-in).

[end of proof]

Thanks for all the feedback.

I'm going to need some time to process everything in the last couple of posts to really integrate all of this.

Hopefully I'll eventually be able to come up with optimal real time "vs range" decisions about whether to ship flop or call and ship/fold turn.

obviously in the scenario we are talking about fold equity is not a consideration